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We derive analytical solutions based on singular Green's functions, whichenable efficient computations of scattering simulations or Floquet-Blochdispersion relations for waves propagating through an elastic plate, whosesurface is patterned by periodic arrays of elastic beams. Our methodology isversatile and allows us to solve a range of problems regarding arrangements ofmultiple beams per primitive cell, over Bragg to deep-subwavelength scales; wecross-verify against finite element numerical simulations to gain furtherconfidence in our approach, which relies upon the hypothesis of Euler-Bernoullibeam theory considerably simplifying continuity conditions such that each beamcan be replaced by point forces and moments applied to the neutral plane of theplate. The representations of Green's functions by Fourier series or Fouriertransforms readily follows, yielding rapid and accurate analytical schemes. Theaccuracy and flexibility of our solutions are demonstrated by engineeringtopologically non-trivial states, from primitive cells with broken spatialsymmetries, following the phononic analogue of the Quantum Valley Hall Effect(QVHE). Topologically protected states are produced and coexist along:interfaces between adjoining chiral-mirrored bulk media and edges between onesuch chiral bulk and the surrounding bare elastic plate, allowing topologicalcircuits to be designed with robust waveguiding; these topologicallynon-trivial states exist within near flexural resonances of the constituentbeams of the phononic crystal, and hence can be tuned into a deep-subwavelengthregime.

Analytical solutions for Bloch waves in resonant phononic crystals: Deep  subwavelength energy splitting and mode steering between topologically  protected interfacial and edge states